# Journal of Operator Theory

Volume 87, Issue 2, Spring 2022  pp. 413-433.

On star moments of the compression of the free unitary Brownian motion by a free projection

Authors:  Nizar Demni (1), Tarek Hamdi (2)
Author institution: (1) Aix-Marseille Universite, CNRS Centrale Marseille, I2M - UMR 7373, 39 rue F. Joliot Curie, 13453 Marseille, France
(2) Department of Management Information Systems, College of Business Management, Qassim University, Ar Rass, Saudi Arabia \textit{and} Laboratoire d'Analyse Mathematiques et Applications LR11ES11, Universite de Tunis El-Manar, Tunisie

Summary: In this paper, we derive explicit expressions for some classes of $\star$-moments of a free unitary Brownian motion compressed by a free projection, using various methods. While the moments of this nonnormal operator are readily derived through analytical or combinatorial methods, we only succeeded to derive its mixed ones after solving a nonlinear partial differential equation (pde) for their generating function. We shall also give some interest in odd alternating moments. In particular, we derive a linear pde for their generating function which we solve when the rank of the projection equals~${1}/{2}$.

DOI: http://dx.doi.org/10.7900/jot.2020oct06.2319
Keywords: free unitary Brownian motion, selfadjoint projection, mixed moments, alternating moments, noncrossing partitions, Kreweras complement